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Von Neumann and Birkhoff ergodic theorems for negatively curved groups. (Théorèmes ergodiques de von Neumann et de Birkhoff sur les groupes à courbure négative.) (English. French summary) Zbl 1359.37007
Summary: We prove maximal inequalities for concentric ball and spherical shell averages on a general Gromov hyperbolic group, in arbitrary probability preserving actions of the group. Under an additional condition, satisfied for example by all groups acting isometrically and properly discontinuously on \(\operatorname{CAT}(-1)\) spaces, we prove a pointwise ergodic theorem with respect to a sequence of probability measures supported on concentric spherical shells.

37A30 Ergodic theorems, spectral theory, Markov operators
22D40 Ergodic theory on groups
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
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